Let x be a rational number in (113/191 ,184/311). Find x=a/b when is known that a<400.

{ 113/191 < a/b < 184/311 }

I have no elegant mathematical answer to your question. But an answer can relatively easily be found with the help of a little program: since there are only finitely many candiadates for a/b to consider.
For example, I have hacked together (rather hastily) the following Python script:

You did a wonderful job!!! But... I looking for a mathematical solution...

Of course, we are all looking for more elegant and general, in a word "mathematical" ways of answering such questions. On the other hand, for this purpose one would want to state the problem in a more general form, for otherwise the effort that goes into finding a general answer might not get amortized properly...

Another question(that comes with your answer...) is: to prove that there is only one number such this.

It comes down to a question about the correctness of this program. There is a small dark cloud on the horizon for example, because this program uses ordinary floating point arithmetic.
That aside, I think that the program does an exhaustive search (there are only finitely many possibilities in any case), and I am thus quite confident that there are no other solutions to your problem.
Btw: Initially I considered the possibility of searching the Stern?Brocot tree - Wikipedia, the free encyclopedia , but came to the conclusion that even if that could be done, the difference to the much simpler approch that I finally used would mainly be to arrive at a much more complex program (with correspondingly higher probability of such a program containing a bug).